Absence of resonance near the critical line on asymptotically hyperbolic spaces

As a consequence of a result of Cardoso–Vodev, we show that the resolvent of the Laplacian on asymptotically hyperbolic manifolds is analytic in an exponential neighbourhood of the critical line $\{\Re(\lambda )=\frac{n}{2}\}$ . The case of non‐trapping metrics with constant curvature near infinity is also considered: there is a strip $\{\Re(\lambda )>\frac{n}{2}-\varepsilon\}$ with a finite number of resonances.